The Chain Rule презентация

student will be able to form the composition of two functions.
The student will be able to apply the general power rule.
The student will be able to apply the chain rule.

of functions f and g if
m(x) = f [g(x)]
The domain of m is the set of all numbers x such that x is in the domain of g and g(x) is in the domain of f.

of the power rule:

Now we want to generalize this rule so that we can differentiate composite functions of the form [u(x)]n, where u(x) is a differentiable function. Is the power rule still valid if we replace x with a function u(x)?

[u(x)]3 = 8x6. Which of the following is f ’(x)?
(a) 3[u(x)]2 (b) 3[u’(x)]2 (c) 3[u(x)]2 u’(x)

[u(x)]3 = 8x6. Which of the following is f ’(x)?
(a) 3[u(x)]2 (b) 3[u’(x)]2 (c) 3[u(x)]2 u’(x)
We know that f ’(x) = 48x5.
(a) 3[u(x)]2 = 3(2x2)2 = 3(4x4) = 12 x4. This is not correct.
(b) 3[u’(x)]2 = 3(4x)2 = 3(16x2) = 48x2. This is not correct.
(c) 3[u(x)]2 u’(x) = 3[2x2]2(4x) = 3(4x4)(4x) = 48x5. This is the correct choice.

example of the generalized power rule: If u is a function of x, then

For example,

and u = g(x) define the composite function y = f (u) = f [g(x)], then

We have used the generalized power rule to find derivatives of composite functions of the form f (g(x)) where f (u) = un is a power function. But what if f is not a power function? It is a more general rule, the chain rule, that enables us to compute the derivatives of many composite functions of the form f(g(x)).

then y’ = nu n - 1 ⋅ du/dx

If y = ln u, then y’ = 1/u ⋅ du/dx

If y = e u, then y ’ = e u ⋅ du/dx

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